Numerical analysis and efficient implementation of fast collocation methods for fractional Laplacian model on nonuniform grids

Abstract

We propose a fast collocation method based on Krylov subspace iterative solver on general nonuniform grids for the fractional Laplacian problem, in which the fractional operator is presented in a singular integral formulation. The method is proved to be uniquely solvable on general nonuniform grids for $\alpha\in(0,1)$, provided that the sum-of-exponentials (SOE) approximation is sufficiently accurate. In addition, a modified scheme is developed and proved to be uniquely solvable on uniform grids for $\alpha\in(0,2)$. Efficient implementation of the proposed fast collocation schemes based on fast matrix-vector multiplication is carefully discussed, in terms of computational complexity and memory requirement. To further improve computational efficiency, a banded preconditioner is incorporated into the Krylov subspace iterative solver. A rigorous maximum-norm error analysis for $\alpha\in(0,1)$ is presented on specific graded grids, which shows that the convergence order depends on the grading parameter. Numerical experiments validate the predicted convergence and demonstrate the efficiency of the fast collocation schemes.